No coordinate system required, that would be Wolf0D.

Sure you would, to determine how far into the level you were. I wasn't talking about changing the level, just putting it in a first person view.

"How far" implies length, something which implicitly requires another dimension, at least "another" when considered from the perspective of a "viewer" in 1D space.

How does "in front of you" exist in 1d?

It would exist. If something exists, it's in front of you. If something exists, it's displayed on the screen with the single pixel you see in the game. I assume you'd differentiate with pixel color, like this game does.